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Theorem trgcopyeu 25743
 Description: Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.)
Hypotheses
Ref Expression
trgcopy.p 𝑃 = (Base‘𝐺)
trgcopy.m = (dist‘𝐺)
trgcopy.i 𝐼 = (Itv‘𝐺)
trgcopy.l 𝐿 = (LineG‘𝐺)
trgcopy.k 𝐾 = (hlG‘𝐺)
trgcopy.g (𝜑𝐺 ∈ TarskiG)
trgcopy.a (𝜑𝐴𝑃)
trgcopy.b (𝜑𝐵𝑃)
trgcopy.c (𝜑𝐶𝑃)
trgcopy.d (𝜑𝐷𝑃)
trgcopy.e (𝜑𝐸𝑃)
trgcopy.f (𝜑𝐹𝑃)
trgcopy.1 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
trgcopy.2 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
trgcopy.3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
Assertion
Ref Expression
trgcopyeu (𝜑 → ∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
Distinct variable groups:   ,𝑓   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝐷,𝑓   𝑓,𝐸   𝑓,𝐹   𝑓,𝐺   𝑓,𝐼   𝑓,𝐿   𝑃,𝑓   𝜑,𝑓   𝑓,𝐾

Proof of Theorem trgcopyeu
Dummy variables 𝑎 𝑏 𝑘 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trgcopy.p . . 3 𝑃 = (Base‘𝐺)
2 trgcopy.m . . 3 = (dist‘𝐺)
3 trgcopy.i . . 3 𝐼 = (Itv‘𝐺)
4 trgcopy.l . . 3 𝐿 = (LineG‘𝐺)
5 trgcopy.k . . 3 𝐾 = (hlG‘𝐺)
6 trgcopy.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 trgcopy.a . . 3 (𝜑𝐴𝑃)
8 trgcopy.b . . 3 (𝜑𝐵𝑃)
9 trgcopy.c . . 3 (𝜑𝐶𝑃)
10 trgcopy.d . . 3 (𝜑𝐷𝑃)
11 trgcopy.e . . 3 (𝜑𝐸𝑃)
12 trgcopy.f . . 3 (𝜑𝐹𝑃)
13 trgcopy.1 . . 3 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
14 trgcopy.2 . . 3 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
15 trgcopy.3 . . 3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15trgcopy 25741 . 2 (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
176ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐺 ∈ TarskiG)
187ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐴𝑃)
198ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐵𝑃)
209ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐶𝑃)
2110ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐷𝑃)
2211ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐸𝑃)
2312ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐹𝑃)
2413ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
2514ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
2615ad5antr 773 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
27 simpl 472 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
2827eleq1d 2715 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
29 simpr 476 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑦 = 𝑏)
3029eleq1d 2715 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
3128, 30anbi12d 747 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸)))))
32 simpr 476 . . . . . . . . . . . 12 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑧 = 𝑡)
33 simpll 805 . . . . . . . . . . . . 13 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑥 = 𝑎)
34 simplr 807 . . . . . . . . . . . . 13 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑦 = 𝑏)
3533, 34oveq12d 6708 . . . . . . . . . . . 12 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑥𝐼𝑦) = (𝑎𝐼𝑏))
3632, 35eleq12d 2724 . . . . . . . . . . 11 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
3736cbvrexdva 3208 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦) ↔ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
3831, 37anbi12d 747 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))))
3938cbvopabv 4755 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}
40 simp-5r 826 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓𝑃)
41 simp-4r 824 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑘𝑃)
42 simpllr 815 . . . . . . . . 9 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
4342simpld 474 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩)
44 simplr 807 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩)
4542simprd 478 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
46 simpr 476 . . . . . . . 8 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
471, 2, 3, 4, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 43, 44, 45, 46trgcopyeulem 25742 . . . . . . 7 ((((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓 = 𝑘)
4847anasss 680 . . . . . 6 (((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘)
4948anasss 680 . . . . 5 ((((𝜑𝑓𝑃) ∧ 𝑘𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))) → 𝑓 = 𝑘)
5049ex 449 . . . 4 (((𝜑𝑓𝑃) ∧ 𝑘𝑃) → (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
5150anasss 680 . . 3 ((𝜑 ∧ (𝑓𝑃𝑘𝑃)) → (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
5251ralrimivva 3000 . 2 (𝜑 → ∀𝑓𝑃𝑘𝑃 (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
53 eqidd 2652 . . . . . 6 (𝑓 = 𝑘𝐷 = 𝐷)
54 eqidd 2652 . . . . . 6 (𝑓 = 𝑘𝐸 = 𝐸)
55 id 22 . . . . . 6 (𝑓 = 𝑘𝑓 = 𝑘)
5653, 54, 55s3eqd 13655 . . . . 5 (𝑓 = 𝑘 → ⟨“𝐷𝐸𝑓”⟩ = ⟨“𝐷𝐸𝑘”⟩)
5756breq2d 4697 . . . 4 (𝑓 = 𝑘 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩))
58 breq1 4688 . . . 4 (𝑓 = 𝑘 → (𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
5957, 58anbi12d 747 . . 3 (𝑓 = 𝑘 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)))
6059reu4 3433 . 2 (∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ↔ (∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ ∀𝑓𝑃𝑘𝑃 (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘)))
6116, 52, 60sylanbrc 699 1 (𝜑 → ∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943   ∖ cdif 3604   class class class wbr 4685  {copab 4745  ‘cfv 5926  (class class class)co 6690  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  cgrGccgrg 25450  hlGchlg 25540  hpGchpg 25694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkgld 25396  df-trkg 25397  df-cgrg 25451  df-ismt 25473  df-leg 25523  df-hlg 25541  df-mir 25593  df-rag 25634  df-perpg 25636  df-hpg 25695  df-mid 25711  df-lmi 25712 This theorem is referenced by: (None)
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